Communications in Mathematical Physics

, Volume 143, Issue 3, pp 431–449 | Cite as

Dual polygonal billiards and necklace dynamics

  • Eugene Gutkin
  • Nandor Simanyi


We study the orbits of the dual billiard map about a polygonal table using the technique of necklace dynamics. Our main result is that for a certain class of tables, called the quasi-rational polygons, the dual billiard orbits are bounded. This implies that for the subset of rational tables (i.e. polygons with rational vertices) the dual billiard orbits are periodic.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [D] Douady, R.: Thèse de 3-ème cycle, Université de Paris 7, 1982Google Scholar
  2. [G1] Gutkin, E.: Billiards on almost integrable polyhedral surfaces. Ergod. Theor. Dynam. Sys.4, 529–584 (1984)Google Scholar
  3. [G2] Gutkin, E.: Billiards in polygons. Physica19D, 311–333 (1986)Google Scholar
  4. [GS] Gutkin, E., Simanyi, N.: Dual billiard about the regular octagon (in preparation)Google Scholar
  5. [K] Kolodziej, R.: The antibilliard outside a polygon. Bull. Poslih Acad. Sci. Math.37, 163–168 (1989)Google Scholar
  6. [M1] J. Moser: Stable and random motions in dynamical systems. Ann. math. Stud., vol. 77. Princeton, NJ: Princeton University Press 1973Google Scholar
  7. [M2] Moser, J.: Is the solar system stable? Mathematical Intelligencer1, 65–71 (1978)Google Scholar
  8. [VS] Vivaldi, F., Shaidenko, A.: Global stability of a class of discontinuous dual billiards. Commun. Math. Phys.110, 625–640 (1987)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1992

Authors and Affiliations

  • Eugene Gutkin
    • 1
  • Nandor Simanyi
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos AngelesUSA

Personalised recommendations